📚 Handling Data 7: Cumulative Frequency, Box Plots and Histograms | 数据处理7：累积频率、箱线图和直方图
Handling Data 7 in Edexcel IGCSE Mathematics builds on earlier statistics and probability topics, introducing students to more sophisticated graphical representations of data along with tools for comparing and analysing distributions. This chapter covers cumulative frequency curves, box plots, histograms with unequal class widths, scatter graphs, and further probability including tree diagrams. Mastering these skills provides a solid foundation for making sense of real-world data and for achieving high marks on the exam.
在 Edexcel IGCSE 数学中，数据处理7建立在早期统计和概率课题的基础上，向学生介绍更复杂的数据图形表示方法，以及比较和分析分布的工具。本章涵盖累积频率曲线、箱线图、组距不等的直方图、散点图，以及包括树状图在内的进一步概率内容。掌握这些技能能为理解现实世界中的数据和获得考试高分打下坚实基础。
1. Cumulative Frequency Tables and Curves | 累积频率表与曲线
A cumulative frequency table is a table that keeps a running total of the frequencies. You add up the frequencies as you go down the table, so each entry shows the total number of data values up to and including that class or group.
累积频率表是一种保持频率运行总和的表格。在表中向下累加各组的频率，每一条记录显示截至该组的上限为止（含该组）所有数据值的总数。
The final cumulative frequency always equals the total number of observations (n). To plot a cumulative frequency curve, we place the upper class boundaries on the horizontal axis and the cumulative frequencies on the vertical axis, then connect the points with a smooth curve.
最后的累积频率始终等于总观测数 (n)。绘制累积频率曲线时，横轴为各组的上边界，纵轴为累积频率，然后将各点用平滑曲线连接起来。
Cumulative frequency diagrams are especially useful for estimating medians, quartiles and percentiles without needing the raw data, as we can simply read values off the curve.
累积频率图在无需原始数据的情况下特别适用于估计中位数、四分位数和百分位数，只需从曲线上读取相应值即可。
2. Finding the Median and Quartiles from a Cumulative Frequency Graph | 从累积频率图中求中位数和四分位数
To find the median from a cumulative frequency graph, first locate half the total frequency (n/2) on the vertical axis. Draw a horizontal line from this point to the curve, then drop a vertical line to the horizontal axis – the reading is the median.
要从累积频率图中求中位数，先在纵轴上找到总频率的一半 (n/2)，从此点向曲线画一条水平线，再从交点向横轴画一条垂直线，该读数即为中位数。
e upper quartile Q₃ at 3n/4 using the same construction. The interquartile range (IQR = Q₃ – Q₁) can then be calculated, which measures the spread of the middle 50% of data.
下四分位数 Q₁ 在 n/4 处找到，上四分位数 Q₃ 在 3n/4 处用同样的作图方法求得。然后可计算出四分位距 (IQR = Q₃ – Q₁)，它衡量中间 50% 数据的离散程度。
If the original data are grouped, these readings give accurate estimates of the median and quartiles, making cumulative frequency curves a powerful data-handling tool.
如果原始数据是分组的，这些读数为中位数和四分位数提供了精确的估计，使累积频率曲线成为一个强大的数据处理工具。
3. Box Plots (Box-and-Whisker Diagrams) | 箱线图（盒须图）
A box plot displays the five-number summary of a dataset: minimum value, lower quartile (Q₁), median, upper quartile (Q₃) and maximum value. The box represents the interquartile range, with a line inside marking the median, and the 'whiskers' extend to the minimum and maximum.
箱线图显示数据集的五数总结：最小值、下四分位数 (Q₁)、中位数、上四分位数 (Q₃) 和最大值。盒子代表四分位距，内部有一条线标示中位数，“须线”延伸到最小值和最大值。
The length of the box and the position of the median inside it tell us about the symmetry or skewness of the distribution. A median near the centre of the box suggests a roughly symmetric distribution, while a median close to Q₁ indicates a positive skew, and near Q₃ indicates a negative skew.
盒子的长度和中位数在盒内的位置告知我们分布的对称性或偏态。中位数靠近盒子中心表明分布大致对称，中位数靠近 Q₁ 表示正偏态，靠近 Q₃ 表示负偏态。
Box plots are excellent for comparing two or more distributions side by side using a common scale, clearly showing differences in central tendency and variability.
箱线图非常适合在同一尺度上并排比较两个或多个分布，可以清晰地显示集中趋势和变异性的差异。
IQR = Q₃ – Q₁
四分位距 = Q₃ – Q₁
4. Drawing Histograms with Unequal Class Widths | 绘制直方图（组距不等的频数分布）
When class widths are unequal, we cannot simply plot frequencies on the vertical axis, because wider classes would be visually overemphasised. Instead, we calculate frequency density using the formula:
当组距不相等时，我们不能简单地在纵轴上绘制频数，因为较宽的组会在视觉上被过度强调。此时我们改用 频率密度 计算，公式为：
Frequency density = Frequency ÷ Class width
频率密度 = 频数 ÷ 组距
In a histogram, the vertical axis represents frequency density and the horizontal axis shows the class boundaries. The area of each bar is proportional to the frequency, so the total area of all bars corresponds to the total frequency.
在直方图中，纵轴表示频率密度，横轴表示组边界。每个条形块的面积与频数成正比，因此所有条形块的总面积对应总频数。
Always check that your frequency density scale is correct and label axes clearly. This representation is required whenever class widths differ, and it is a common exam topic.
务必检查频率密度刻度是否正确并清晰标注数轴。只要组距不同，就必须使用这种表示方法，这也是常见的考试课题。
5. Frequency Polygons | 频率多边形
A frequency polygon is a line graph that joins the midpoints of the tops of the histogram bars (or the midpoints of grouped data plotted at their respective frequencies). It can also be drawn independently of the histogram.
频率多边形是一种折线图，连接直方图各条形顶部的中点（或分组数据中点对应的频数）。它也可以独立于直方图进行绘制。
To construct a frequency polygon, find the midpoint of each class interval, plot the points (midpoint, frequency), and join them with straight lines. It is common to add extra zeros at the start and end so the line meets the horizontal axis.
要绘制频率多边形，先找出每个组区间的中点，以（中点，频数）描点，然后用直线连接。通常会在首尾加上额外的零值点，使折线最终与横轴相交。
Frequency polygons are especially useful for comparing two frequency distributions on the same graph, as overlapping outlines are easier to read than multiple histograms.
频率多边形特别适用于在同一图形上比较两个频数分布，因为重叠的轮廓线比多个直方图更容易读出差异。
6. Scatter Graphs and Correlation | 散点图与相关性
A scatter graph (or scatter plot) is used to display the relationship between two variables. Each point represents a pair of values (x, y), and the overall pattern indicates the type of correlation: positive, negative, or no correlation.
散点图（散点图）用于展示两个变量之间的关系。每个点代表一对数值 (x, y)，整体图样揭示了相关类型：正相关、负相关或无相关。
Positive correlation means that as x increases, y tends to increase; negative correlation means as x increases, y tends to decrease. If no clear pattern is visible, we say there is no correlation.
正相关意味着 x 增加时 y 也趋于增加；负相关意味着 x 增加时 y 趋于减少。如果看不到明显图样，则称无相关。
A line of best fit (a straight line drawn through the points to balance those above and below) can be used to estimate unknown values. Remember that correlation does not imply causation – a strong correlation may be coincidental or due to a third factor.
最佳拟合线（穿过各点，使线上方和下方的点数大致平衡的直线）可用于估计未知值。请记住，相关性并不意味着因果关系——高度相关可能只是巧合，或由第三个因素所致。
y ≈ mx + c (line of best fit)
y ≈ mx + c （最佳拟合线）
7. Probability Basics and Relative Frequency | 概率基础与相对频率
The probability of an event A, written P(A), is given by the ratio of the number of favourable outcomes to the total number of equally likely outcomes:
事件 A 的概率，记作 P(A)，等于有利结果数除以所有等可能的结果总数：
P(A) = Number of favourable outcomes ÷ Total number of outcomes
P(A) = 有利结果数 ÷ 总结果数
Probabilities always lie between 0 and 1 inclusive. When outcomes are mutually exclusive – meaning they cannot happen at the same time – the probability that either A or B occurs is P(A) + P(B).
概率值始终介于 0 到 1 之间（含）。若结果互斥——意味着它们不能同时发生——则 A 或 B 发生的概率为 P(A) + P(B)。
times the event occurs) ÷ (total number of trials). As the number of trials increases, relative frequency tends to settle down to the theoretical probability.
相对频率是基于实验数据的概率估计值：相对频率 = （事件发生次数）÷（总试验次数）。随着试验次数增加，相对频率会趋于稳定到理论概率。
8. Probability Tree Diagrams | 概率树状图
Tree diagrams are used to list all possible outcomes of two or more sequential events. Each branch is labelled with a probability; at each stage the probabilities on the branches must sum to 1.
树状图用于列出两个及以上连续事件所有可能的结果。每条分支标注一个概率；在每个阶段各分支的概率总和必须为 1。
To find the probability of a particular combined outcome, multiply the probabilities along the successive branches that lead to that outcome. If more than one path gives the same final outcome, add the resulting probabilities.
要求某一特定组合结果的概率，将通向该结果的连续分支上的概率相乘。如果有多条路径得到相同的最终结果，则将这些概率相加。
Tree diagrams can handle both independent events and, with careful adjustment of second-stage probabilities, conditional events such as 'without replacement' situations. Always check whether the first selection affects the sample space for the second.
树状图既可以处理独立事件，也可以通过谨慎调整第二阶段概率来处理条件事件，例如“不放回”情形。务必检查第一次抽取是否影响第二次的样本空间。
9. Conditional Probability | 条件概率
Conditional probability is the probability of an event A occurring given that event B has already occurred, written as P(A|B). It is calculated by:
条件概率是指在事件 B 已发生的前提下事件 A 发生的概率，记作 P(A|B)。计算公式为：
P(A|B) = P(A ∩ B) ÷ P(B)
P(A|B) = P(A ∩ B) ÷ P(B)
In the context of tree diagrams, when events are not replaced, the probabilities on the second set of branches change depending on what happened in the first event. This makes tree diagrams a natural tool for visualising conditional probabilities.
在树状图的语境中，当事件不放回时，第二组分支上的概率会根据第一次事件的结果而改变。这使得树状图成为可视化条件概率的天然工具。
Understanding conditional probability is essential for solving problems involving successive dependent events and for interpreting real-world scenarios where outcomes are linked.
理解条件概率对于解决涉及连续相关事件的问题以及解读结果相互关联的现实情景至关重要。
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